Bsi bs en 62226 2 1 2005

7 2 Analytical models ...7 2.1 General ...8 2.2 Basic analytical models for uniform fields ...7 3 Numerical models ...9 3.1 General information about numerical models ...9 3.2 2D models

Exposure to electric or

magnetic fields in the

low and intermediate

frequency range —

Methods for calculating

the current density and

internal electric field

induced in the human

This British Standard was

published under the authority

of the Standards Policy and

This British Standard is the official English language version of

EN 62226 -2-1:2005 It is identical with IEC 62226-2-1:2004

The UK participation in its preparation was entrusted to Technical Committee GEL/106, Human exposure to Lf and Hf Electromagnetic radiation, which has the responsibility to:

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Amendments issued since publication

EUROPÄISCHE NORM January 2005

CENELEC

Central Secretariat: rue de Stassart 35, B - 1050 Brussels

© 2005 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members

Ref No EN 62226-2-1:2005 E

ICS 17.220.20

English version

Exposure to electric or magnetic fields

in the low and intermediate frequency range – Methods for calculating the current density and internal electric field induced in the human body

Part 2-1: Exposure to magnetic fields –

Méthodes de calcul des densités

de courant induit et des champs

électriques induits dans le corps humain

Partie 2-1: Exposition à des champs

Verfahren zur Berechnung der induzierten Körperstromdichte und des im

menschlichen Körper induzierten elektrischen Feldes

Teil 2-1: Exposition gegenüber magnetischen Feldern – 2D-Modelle

(IEC 62226-2-1:2004)

This European Standard was approved by CENELEC on 2004-12-01 CENELEC members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration

Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the Central Secretariat or to any CENELEC member

This European Standard exists in three official versions (English, French, German) A version in any other language made by translation under the responsibility of a CENELEC member into its own language and notified to the Central Secretariat has the same status as the official versions

CENELEC members are the national electrotechnical committees of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom

Foreword

The text of document 106/79/FDIS, future edition 1 of IEC 62226-2-1, prepared by IEC TC 106, Methods for the assessment of electric, magnetic and electromagnetic fields associated with human exposure, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as

EN 62226-2-1 on 2004-12-01

This Part 2-1 is to be used in conjunction with EN 62226-11)

The following dates were fixed:

– latest date by which the EN has to be implemented

at national level by publication of an identical

– latest date by which the national standards conflicting

CONTENTS

FOREWORD 9

INTRODUCTION 6

1 Scope 7

2 Analytical models 7

2.1 General 8

2.2 Basic analytical models for uniform fields 7

3 Numerical models 9

3.1 General information about numerical models 9

3.2 2D models – General approach 10

3.3 Conductivity of living tissues 11

3.4 2D Models – Computation conditions 12

3.5 Coupling factor for non-uniform magnetic field 12

3.6 2D Models – Computation results 13

4 Validation of models 15

Annex A (normative) Disk in a uniform field 16

Annex B (normative) Disk in a field created by an infinitely long wire 19

Annex C (normative) Disk in a field created by 2 parallel wires with balanced currents 27

Annex D (normative) Disk in a magnetic field created by a circular coil 38

Annex E (informative) Simplified approach of electromagnetic phenomena 50

Annex F (informative) Analytical calculation of magnetic field created by simple induction systems: 1 wire, 2 parallel wires with balanced currents and 1 circular coil 52

Annex G (informative) Equation and numerical modelling of electromagnetic phenomena for a typical structure: conductive disk in electromagnetic field 54

Bibliography 56

Figure 1 – Conducting disk in a uniform magnetic flux density 8

Figure 2 – Finite elements meshing (2nd order triangles) of a disk, and detail 10

Figure 3 – Conducting disk in a non-uniform magnetic flux density 11

Figure 4 – Variation with distance to the source of the coupling factor for non-uniform magnetic field, K, for the three magnetic field sources (disk radius R = 100 mm) 14

Figure A.1 – Current density lines J and distribution of J in the disk 16

Figure A.2 – J = f [r]: Spot distribution of induced current density calculated along a diameter of a homogeneous disk in a uniform magnetic field 17

Figure A.3 – Ji = f [r]: Distribution of integrated induced current density calculated along a diameter of a homogeneous disk in a uniform magnetic field 18

Figure B.1 – Disk in the magnetic field created by an infinitely straight wire 19

Figure B.2 – Current density lines J and distribution of J in the disk (source: 1 wire, located at d = 10 mm from the edge of the disk) 20

Figure B.3 – Spot distribution of induced current density along the diameter AA of the

disk (source: 1 wire, located at d = 10 mm from the edge of the disk) 20 Figure B.4 – Distribution of integrated induced current density along the diameter AA

of the disk (source: 1 wire, located at d = 10 mm from the edge of the disk) 21

Figure B.5 – Current density lines J and distribution of J in the disk (source: 1 wire,

located at d = 100 mm from the edge of the disk) 21Figure B.6 – Distribution of integrated induced current density along the diameter AA

of the disk (source: 1 wire, located at d = 100 mm from the edge of the disk) 22

Figure B.7 – Parametric curve of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm) 23

Figure B.8 – Parametric curve of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm) 24

Figure B.9 – Parametric curve of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm) 25

Figure B.10 – Parametric curve of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm) 26 Figure C.1 – Conductive disk in the magnetic field generated by 2 parallel wires with

balanced currents 27

Figure C.2 – Current density lines J and distribution of J in the disk (source: 2 parallel

wires with balanced currents, separated by 5 mm, located at d = 7,5 mm from the

edge of the disk) 28

Figure C.3 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk (source: 2 parallel wires with balanced currents,

separated by 5 mm, located at d = 7,5 mm from the edge of the disk) 28

Figure C.4– Current density lines J and distribution of J in the disk (source: 2 parallel

wires with balanced currents separated by 5 mm, located at d = 97,5 mm from the

edge of the disk) 29

Figure C.5 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk (source: 2 parallel wires with balanced currents

separated by 5 mm, located at d = 97,5 mm from the edge of the disk) 29

Figure C.6 – Parametric curves of factor K for distances up to 300 mm to a source

consisting of 2 parallel wires with balanced currents and for different distances e

between the 2 wires (homogeneous disk R = 100 mm) 30

Figure C.7 – Parametric curves of factor K for distances up to 1 900 mm to a source

consisting of 2 parallel wires with balanced currents and for different distances e

between the 2 wires (homogeneous disk R = 100 mm) 32

Figure C.8 – Parametric curves of factor K for distances up to 300 mm to a source

consisting of 2 parallel wires with balanced currents and for different distances e

between the 2 wires (homogeneous disk R = 200 mm) 34

Figure C.9 – Parametric curves of factor K for distances up to 1 900 mm to a source

consisting of 2 parallel wires with balanced currents and for different distances e

between the 2 wires (homogeneous disk R = 200 mm) 36 Figure D.1 – Conductive disk in a magnetic field created by a coil 38

Figure D.2 –Current density lines J and distribution of J in the disk (source: coil of

radius r = 50 mm, conductive disk R = 100 mm, d = 5 mm) 39

Figure D.3 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk (source: coil of radius r = 50 mm, conductive disk

R = 100 mm, d = 5 mm) 39

Figure D.4 – Current density lines J and distribution of J in the disk (source: coil of

radius r = 200 mm, conductive disk R = 100 mm, d = 5 mm) 40

Figure D.5 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk (source: coil of radius r = 200 mm, conductive disk

R = 100 mm, d = 5 mm) 40

Figure D.6 – Current density lines J and distribution of J in the disk (source: coil of

radius r = 10 mm, conductive disk R = 100 mm, d = 5 mm) 41

Figure D.7 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk (source: coil of radius r = 10 mm, conductive disk

R = 100 mm, d = 5 mm) 41

Figure D 8 – Parametric curves of factor K for distances up to 300 mm to a source

consisting of a coil and for different coil radius r (homogeneous disk R = 100 mm) 42

Figure D.9 – Parametric curves of factor K for distances up to 1 900 mm to a source

consisting of a coil and for different coil radius r (homogeneous disk R = 100 mm) 44

Figure D.10 – Parametric curves of factor K for distances up to 300 mm to a source

consisting of a coil and for different coil radius r (homogeneous disk R = 200 mm) 46

Figure D.11 – Parametric curves of factor K for distances up to 1 900 mm to a source

consisting of a coil and for different coil radius r (homogeneous disk R = 200 mm) 48

Table 1 – Numerical values of the coupling factor for non-uniform magnetic field K for

different types of magnetic field sources, and different distances between sources and

conductive disk (R = 100 mm) 15

Table B.1 – Numerical values of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm) 23

Table B.2 –Numerical values of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm) 24

Table B.3 – Numerical values of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm) 25

Table B.4 –Numerical values of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm) 26

Table C.1 – Numerical values of factor K for distances up to 300 mm to a source

consisting of 2 parallel wires with balanced currents (homogeneous disk: R = 100 mm) 31

Table C.2 – Numerical values of factor K for distances up to 1 900 mm to a source

consisting of 2 parallel wires with balanced currents (homogeneous disk: R = 100 mm) 33

Table C.3 – Numerical values of factor K for distances up to 300 mm to a source

consisting of 2 parallel wires with balanced currents (homogeneous disk: R = 200 mm) 35

Table C.4 – Numerical values of factor K for distances up to 1 900 mm to a source

consisting of 2 parallel wires with balanced currents (homogeneous disk: R = 200 mm) 37

Table D.1 – Numerical values of factor K for distances up to 300 mm to a source

consisting of a coil (homogeneous disk: R = 100 mm) 43

Table D.2 – Numerical values of factor K for distances up to 1 900 mm to a source

consisting of a coil (homogeneous disk: R = 100 mm) 45

Table D.3 – Numerical values of factor K for distances up to 300 mm to a source

consisting of a coil (homogeneous disk: R = 200 mm) 47

Table D.4 – Numerical values of factor K for distances up to 1 900 mm to a source

consisting of a coil (homogeneous disk: R = 200 mm) 49

INTRODUCTION

Public interest concerning human exposure to electric and magnetic fields has led international and national organisations to propose limits based on recognised adverse effects

This standard applies to the frequency range for which the exposure limits are based on the induction of voltages or currents in the human body, when exposed to electric and magnetic fields This frequency range covers the low and intermediate frequencies, up to 100 kHz Some methods described in this standard can be used at higher frequencies under specific conditions

The exposure limits based on biological and medical experimentation about these fundamental induction phenomena are usually called “basic restrictions” They include safety factors

The induced electrical quantities are not directly measurable, so simplified derived limits are also proposed These limits, called “reference levels”, are given in terms of external electric and magnetic fields They are based on very simple models of coupling between external fields and the body These derived limits are conservative

Sophisticated models for calculating induced currents in the body have been used and are the subject of a number of scientific publications These use numerical 3D electromagnetic field computation codes and detailed models of the internal structure with specific electrical characteristics of each tissue within the body However such models are still developing; the electrical conductivity data available at present has considerable shortcomings; and the spatial resolution of models is still advancing Such models are therefore still considered to be

in the field of scientific research and at present it is not considered that the results obtained from such models should be fixed indefinitely within standards However it is recognised that such models can and do make a useful contribution to the standardisation process, specially for product standards where particular cases of exposure are considered When results from such models are used in standards, the results should be reviewed from time to time to ensure they continue to reflect the current status of the science

EXPOSURE TO ELECTRIC OR MAGNETIC FIELDS

IN THE LOW AND INTERMEDIATE FREQUENCY RANGE –

METHODS FOR CALCULATING THE CURRENT DENSITY

AND INTERNAL ELECTRIC FIELD INDUCED IN THE HUMAN BODY –

Part 2-1: Exposure to magnetic fields –

2D models

1 Scope

This part of IEC 62226 introduces the coupling factor K, to enable exposure assessment for

complex exposure situations, such as non-uniform magnetic field or perturbed electric field

The coupling factor K has different physical interpretations depending on whether it relates to

electric or magnetic field exposure

The aim of this part is to define in more detail this coupling factor K, for the case of simple

models of the human body, exposed to non-uniform magnetic fields It is thus called “coupling

factor for non-uniform magnetic field”

All the calculations developed in this document use the low frequency approximation in which

displacement currents are neglected This approximation has been validated in the low

frequency range in the human body where parameter εω <<σ

For frequencies up to a few kHz, the ratio of conductivity and permittivity should be calculated

to validate this hypothesis

2.1 General

Basic restrictions in guidelines on human exposure to magnetic fields up to about 100 kHz are

generally expressed in terms of induced current density or internal electric field These

electrical quantities cannot be measured directly and the purpose of this document is to give

methods and tools on how to assess these quantities from the external magnetic field

The induced current density J and the internal electric field Ei are closely linked by the simple

relation:

where σ is the conductivity of living tissues

For simplicity, the content of this standard is presented in terms of induced current densities

J, from which values of the internal electric field can be easily derived using the previous

formula

Analytical models have been used in EMF health guidelines to quantify the relationship

between induced currents or internal electric field and the external fields These involve

assumptions of highly simplified body geometry, with homogeneous conductivity and uniform

applied magnetic field Such models have serious limitations The human body is a much

more complicated non-homogeneous structure, and the applied field is generally non-uniform

because it arises from currents flowing through complex sets of conductors and coils

For example, in an induction heating system, the magnetic field is in fact the superposition of

an excitation field (created by the coils), and a reaction field (created by the induced currents

in the piece) In the body, this reaction field is negligible and can be ignored

Annex E and F presents the analytical calculation of magnetic field H created by simple

sources and Annex G presents the analytical method for calculating the induced current in a

conductive disk

2.2 Basic analytical models for uniform fields

The simplest analytical models used in EMF health guidelines are based on the hypothesis of

coupling between a uniform external magnetic field at a single frequency, and a homogeneous

disk of given conductivity, used to represent the part of the body under consideration, as

illustrated in Figure 1 Such models are used for example in the ICNIRP 1) and NRPB 2)

Figure 1 – Conducting disk in a uniform magnetic flux density

The objective of such a modelling is to provide a simple method to assess induced currents

and internal fields This very first approach is simple and gives conservative values of the

electrical quantities calculated

For alternating magnetic fields, the calculation assumes that the body or the part of the body

exposed is a circular section of radius r, with conductivity σ The calculation is made under

maximum coupling conditions i.e with a uniform magnetic field perpendicular to this disk In

this case, the induced current density at radius r is given by:

dt

dB r r J

2)

where B is the magnetic flux density

_

1) Health Physics (vol 74, n° 4, April 1998, pp 496-522)

2) NRPB, 1993, Board Statement on Restrictions on Human Exposure to Static and Time-varying Electromagnetic

Fields and Radiation, Volume 4, No 5, 1

For a single frequency f, this becomes:

rfB r

As illustrated in Figure 1 (see also Annex A), induced currents are distributed inside the disk,

following a rotation symmetry around the central axis of the disk The value of induced

currents is minimum (zero) at the centre and maximum at the edge of the disk

3.1 General information about numerical models

Simple models, which take into consideration field characteristics, are more realistic than

those, which consider only uniform fields, such as analytical ones

Electromagnetic fields are governed by Maxwell's equations These equations can be

accurately solved in 2- or 3-dimensional structures (2D or 3D computations) using various

numerical methods, such as:

– finite elements method (FEM);

– boundary integral equations method (BIE or BEM), or moment method;

– finite differences method (FD);

– impedance method (IM)

Others methods derive from these For example, the following derive from the finite

differences method:

– finite difference time domain (FDTD);

– frequency dependent finite difference time domain ((FD)2TD);

– scalar potential finite difference (SPFD)

Hybrid methods have been also developed in order to improve modelling (example: FE + BIE)

Commercially available software can accurately solve Maxwell’s equations by taking into

account real geometrical structures and physical characteristics of materials, as well as in

steady state or transient current source conditions

The choice of the numerical method is guided by a compromise between accuracy,

computational efficiency, memory requirements, and depends on many parameters, such as:

– simulated field exposure;

– size and shape of human object to be modelled;

– description level of the human object (size of voxel), or fineness of the meshing;

– frequency range, in order to neglect some parts of Maxwell’s relations (example:

displacement current term for low frequency);

– electrical supply signal (sinusoidal, periodic or transient);

– type of resolution (2D or 3D);

– mathematical formulation;

– linear or non linear physical parameters (conductivity, …);

– performances of the numerical method;

– etc

Computation times can therefore vary significantly

Computed electromagnetic values can be presented in different ways, including:

– distributions of magnetic field H, flux density B, electric field E, current density J These

distributions can be presented in the form of coloured iso-value lines and/or curves, allowing a visual assessment of the phenomena and the possible "hot" points;

– local or spatial averaged integral values of H, B, E, J, etc.;

– global magnitude values: active power

These methods are very helpful for solving specific problems; however they cannot be conveniently used to study general problems

3.2 2D models – General approach

In order to gain quickly an understanding of induced currents in the human body, 2D simulations can be performed using a simple representation of the body (a conductive disk: example of modelling given in Figure 2) in a non-uniform magnetic field, as illustrated in Figure 3

−100 0 100

Figure 2 – Finite elements meshing (2 nd order triangles) of a disk, and detail

IEC 1550/04

Figure 3 – Conducting disk in a non-uniform magnetic flux density

Starting from Maxwell’s relations (low frequency approximation), a single equation can be

obtained with a specific mathematical formulation (see Annex G):

t

H t

H H

v

0 0

2

where

Hex is the excitation field created by the source currents,

Hr is the reaction field created by the induced currents:

)(HrCurl

Jv v

Equation (4) is solved for a 2D geometry using the finite element method applied to the

meshing illustrated in Figure 2

The excitation field Hex is calculated for three non-uniform field sources using the analytical

expressions given in Annex F The three sources modelled are: a current flowing through an

infinitely long wire, two parallel wires with balanced currents and a current loop

X, Y, Z co-ordinates are used XY-plane is the study plane of the disk in which induced

currents are generated Except for the particular case where Hex is uniform, source currents

are in the same plane Only the one component of Hex along the Z-axis is taken into account

The induced currents in the disk have two components J x , J y

Examples of numerical results are presented in Annexes A to D

3.3 Conductivity of living tissues

The computation of induced currents in the body from the external magnetic field is strongly

affected by the conductivity of the different tissues in the body and their anisotropic

properties The results presented in this document assume that the conductivity is

homogeneous and isotropic with a value of 0,2 S/m This value is consistent with the average

value assumed in EMF health guidelines

The most recent assessment of the available data indicates the average conductivity to be

slightly higher: 0,22 S/m More experimental work is in progress to provide more reliable

conductivity information The preferred average conductivity could be changed in the future as

improved information becomes available In that situation the values of induced current

presented in this report should be revised in proportion to the conductivity Nevertheless, the

coupling factor for non-uniform magnetic field K, defined previously, is independent of the

conductivity

3.4 2D Models – Computation conditions

2D computation codes were used to simulate the current induced in a conductive disk by an

alternating magnetic field of frequency f, produced by four different field sources:

– uniform and unidirectional field in all considered space (Annex A);

– current flowing through one infinitely long wire (Annex B);

– 2 parallel wires with balanced currents (Annex C);

– current flowing through one circular coil (Annex D)

In order to facilitate comparisons with analytical models, all numerical values of computation

parameters are fixed throughout this standard:

– radius of disk: R = 100 mm, and R = 200 mm;

– conductivity of disk: σ = 0,2 S/m;

– field sources at 50 Hz frequency

With the exception of the first of the four field sources, the magnetic field from the source is

non-uniform, decreasing with increasing distance from the source In these cases the field

value quoted is the value at the edge of the disk closest to the source

The reaction field created by the induced current in the disk is negligible (due to the very low

conductivity of the disk) and is ignored

3.5 Coupling factor for non-uniform magnetic field

The current density induced in the disk by a localised source of magnetic field (therefore

generating a non-uniform field), is always lower than the current density that would be

induced by a uniform magnetic field whose magnitude is equal to the magnitude of the

non-uniform field at the edge of the disk closest to the localised source This reduction of induced

current for non-uniform field sources is quantified using the coupling factor for non-uniform

magnetic field K, which is physically defined as:

Jnonuniform is the maximum induced current density in the disk exposed to the non-uniform

magnetic field from a localised source,

Juniform is the maximum induced current density in the disk exposed to a uniform

magnetic field

Juniform is derived from equation (3):

RfB R

r J

It shall be noted that K = 1 when the field is uniform Annex A illustrates the current

distribution in a disk of radius R = 100 mm for an applied uniform field B = 1,25 µT The

coupling factor for uniform magnetic field K is calculated numerically for the three

non-uniform sources of field, in Annex B, C and D respectively

NOTE 1 Calculated spot values of induced current densities have been averaged in this document (see Annexes

A to D) So the values of Juniform and Jnonuniform given here above are averaged values, integrated over a cross

section of 1 cm2, perpendicular to the current direction

NOTE 2 Values of K are calculated at a frequency of 50 Hz Nevertheless, due to the low frequency

approximation, these values are also valid for the whole frequency range covered by this standard i.e up to

100 kHz Also, due to the low frequency approximation, K is independent of the conductivity

For real cases, the spatial arrangement of field cannot easily be described in equations, and

the coupling factor K can only be estimated (for example using the table values given in

annexes of this document)

3.6 2D Models – Computation results

This subclause is a summary of the detailed numerical results given in Annexes B, C and D,

which deal with the three types of sources Whatever the source, the model of human body is

treated as a homogeneous disk:

– conductivity of disk: σ = 0,2 S/m

For comparison between the different types of sources (i.e coupling models), the value of the

local maximum magnetic field is normalised Whatever the source, the magnetic field

magnitude at the edge of the disk closest to the source is equal to the uniform field magnitude

(i.e B = 1,25 µT, see annex A)

Table 1 presents a selection from Annexes B, C and D of the numerical values of the factor K

for the three different sources and for a disk radius R = 100 mm These results are also

presented in a graphic form in Figure 4

All the values in Table 1 are less than 1, and sometimes much less than 1, by a factor up to

about 100 This demonstrates that, for a specified maximum current density in the disk, the

corresponding magnetic field at the edge of the disk can have a wide range of values

depending on the characteristics of the field source and on the distance between the disk and

source

The uniform field approximation (for which K = 1) is appropriate only when the distance

between the source and the “human disk” becomes large relative to the size of the disk

(typically 10 times the disk radius) At more usual distance of exposure from, for example,

domestic appliances, the non-uniformity of the magnetic field with the distance has to be

taken into account in the way presented in this standard

0 0,1

Figure 4 – Variation with distance to the source of the coupling factor for non-uniform

magnetic field, K, for the three magnetic field sources (disk radius R = 100 mm)

Table 1 – Numerical values of the coupling factor for non-uniform magnetic field K for

different types of magnetic field sources, and different distances between sources and

conductive disk (R = 100 mm)

K factor for different sources

Distance between the

source and the disk

The validation of the numerical tools used for computation of induced current densities shall

be made by comparison with the results given in the annexes of this standard, which have

been validated by comparison with scientific literature

Additional information concerning the software used for the validation of numerical

computation can be found in the bibliographic references of IEC 62226-1

Annex A

(normative)

Disk in a uniform field

The induced currents are calculated in a disk of homogeneous conductivity In order to allow comparison between different field sources configurations (depending on geometry of the source and distance to the disk, see Annex B to D) the following standard values have been chosen:

– f, frequency = 50 Hz (see note 2 in 3.5);

– B, uniform magnetic flux density = 1,25 µT;

– R, radius of the conductive disk = 100 mm;

– σ, conductivity (homogeneous) = 0,2 S/m

Using these values in equation (3) gives, at the edge of the disk:

Jmax = 0,393 × 10–5 A/m2 (analytical calculation)

Results of a numerical computation using finite element methods are presented hereafter in the form of graphs giving the shape of the distribution of induced currents in the disk (Figure A.1) and curve giving numerical values of local induced currents (Figure A.2):

Figure A.1 – Current density lines J and distribution of J in the disk

This computation gives, at the edge of the disk a value of:

Jmax = 0,390 × 10–5 A/m2

Considering the meshing effect of numerical models, this numerical value of Jmax can be

considered as equal to the analytical one So, analytical and numerical approaches give very

similar results in this simple case

IEC 1553/04

The induced current density varies linearly with distance along a diameter of the disk as

shown in Figure A.2:

0 1,0

Figure A.2 – J = f [r]: Spot distribution of induced current density calculated along a

diameter of a homogeneous disk in a uniform magnetic field

To avoid any bias due to numerical meshing, calculated spot values shall be averaged In the

computations of the present document, a square section of 1 cm2, perpendicular to the

current direction was used

The corresponding analytical formula is the integral of equation (3):

2 /

/1)(

m

m

r r r r m

where r m is the length of integration, equal to 1 cm (valid for r < R – r m/2)

Using the numerical values previously defined, the analytical solution of equation (A-1) is:

J i max = 0,375 × 10–5 A/m2

which is very similar to the numerical value: J i max = 0,374 × 10–5 A/m2

Due to the integration, this value is lower than the spot value

The distribution of the integrated induced current density is also a linear function of the

position of calculation point along a diameter of the disk, as illustrated in Figure A.3:

0 1,0 2,0 3,0 4,0

Figure A.3 – J i = f [r]: Distribution of integrated induced current density calculated

along a diameter of a homogeneous disk in a uniform magnetic field

Annex B

(normative)

Disk in a field created by an infinitely long wire

The induced currents are calculated in a disk of homogeneous conductivity In order to allow

comparison between different field sources configurations (depending on geometry of the

source and distance to the disk) the following standard values have been chosen:

– f, frequency = 50 Hz (see note 2 of 3.5);

– B, magnetic flux density = 1,25 µT, at the edge of the disk closer to the field source;

– R, radius of the conductive disk = 100 mm or 200 mm;

– σ, conductivity (homogeneous) = 0,2 S/m

In this annex, the field source is an alternating current flowing through an infinite straight wire

The conductive disk and the field source are located in the same plane, at a distance d (see

Figure B.1 – Disk in the magnetic field created by an infinitely straight wire

The distance d is the minimum distance between the edge of the disk and the closer part of

the source

The variation of the coupling factor for non-uniform magnetic field K is studied with regard to

the distance d for:

– exposure close to the source: 0 < d < 300 mm

– exposure at higher distance: 0 < d < 1 900 mm

For illustrations and examples of induced currents computation, 3 distances d have been

studied:

– d = 10 mm;

– d = 100 mm;

– d = 1 000 mm

B.1 Calculations for a conductive disk with a radius R = 100 mm

B.1.1 Examples of calculation of inducted currents in the disk

B.1.1.1 Distance to the source d = 10 mm

Results of the computation of local induced currents in the disk are given hereunder in form of graphs giving the shape of the distribution of induced currents in the disk (Figure B.2) and curves giving numerical values of the induced currents (Figures B.3 and B.4) The curve in Figure B.4 gives the distribution of the induced currents integrated over a surface of 1 cm2perpendicular to the induced current direction

Figure B.2 – Current density lines J and distribution of J in the disk

(source: 1 wire, located at d = 10 mm from the edge of the disk)

0 0,2 0,4 0,6 0,8 1,0 1,2

Figure B.3 – Spot distribution of induced current density along the diameter AA of the

disk (source: 1 wire, located at d = 10 mm from the edge of the disk)

NOTE The diameter AA is located as illustrated in Figures B.1 and B.2

Diameter AA

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0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Figure B.4 – Distribution of integrated induced current density along the diameter AA of

the disk (source: 1 wire, located at d = 10 mm from the edge of the disk)

B.1.1.2 Distance to the source d = 100 mm

Results of the computation of local induced currents in the disk are given hereunder in form of

graphs giving the shape of the distribution of induced currents in the disk (Figure B.5) The

curve in Figure B.6 gives the numerical values of the distribution of the induced currents

integrated over a surface of 1 cm2 perpendicular to the induced current direction

Figure B.5 – Current density lines J and distribution of J in the disk

(source: 1 wire, located at d = 100 mm from the edge of the disk)

Diameter AA

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0 1,0 2,0 3,0

Figure B.6 – Distribution of integrated induced current density along the diameter AA

of the disk (source: 1 wire, located at d = 100 mm from the edge of the disk)

B.1.1.3 Distance to the source d = 1 000 mm

Current density lines J, distribution of J in the disk and distribution of induced current density

calculated on the diameter of the disk are similar to those computed in the case of a uniform field (Annex A)

The higher is the distance d between the source and the disk, the lower is the difference with

the computation results obtained with the hypothesis of uniform field: in the present case,

J i max = 0,353 × 10–5 A/m2, to be compared to the value calculated with a uniform field

(J i max = 0,375 × 10–5 A/m2)

B.1.2 Calculated values of the coupling factor for non-uniform magnetic field K

Results of the computation of the coupling factor for non-uniform magnetic field K, as a function of the distance d, are given hereunder in a graphic form (see Figures B.7 and B.8)

Corresponding numerical values are given in Tables B.1 and B.2

The distance d is the minimum distance between the edge of the disk and the closer part of

the source

B.1.2.1 Calculations for short distances to the source: 0 < d < 300 mm

Figure B.7 – Parametric curve of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire

(disk: R = 100 mm)

Table B.1 – Numerical values of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire

B.1.2.2 Calculations for higher distances: 0 < d < 1 900 mm

0 0,1

Figure B.8 – Parametric curve of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm)

Table B.2 – Numerical values of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 100 mm)

Distance between the

source and the disk

Distance between the

source and the disk

B.2 Calculations for a conductive disk with a radius R = 200 mm

Results of the computation of the coupling factor for non-uniform magnetic fields K , as a

function of the distance d, are given hereunder in a graphic form (see Figures B.9 and B.10)

Corresponding numerical values are given in Tables B.3 and B.4

The distance d is the minimum distance between the edge of the disk and the closest part of

Figure B.9 – Parametric curve of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm)

Table B.3 – Numerical values of factor K for distances up to 300 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm)

Distance between the

source and the disk

Distance between the

source and the disk

Distance between the

source and the disk

B.2.2 Calculations for higher distances to the source : 0 < d < 1 900 mm

Figure B.10 – Parametric curve of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm)

Table B.4 –Numerical values of factor K for distances up to 1 900 mm to a source

consisting of an infinitely long wire (disk: R = 200 mm)

Annex C

(normative)

Disk in a field created by 2 parallel wires

with balanced currents

The induced currents are calculated in a disk of homogeneous conductivity In order to allow

comparison between different field sources configurations (depending on the geometry of the

source and distance to the disk) the following standard values have been chosen:

– f, frequency = 50 Hz (see Note 2 in 3.5);

– B, magnetic flux density = 1,25 µT , at the edge of the disk closer to the field source;

– R, radius of the conductive disk = 100 mm and 200 mm;

– σ, conductivity (homogeneous) = 0,2 S/m

In this annex, the magnetic field is generated by a set of 2 parallel wires with balanced

currents (these straight and infinitely long wires are a simplified representation of an electrical

transmission or distribution line) The conductive disk and the field source are located in the

same plane, at a distance d , and the 2 wires are separated by a distance e (see Figure C.1)

The evolution of the coupling factor for non-uniform magnetic field K is studied with regard to

the distance d for:

– exposure close to the source: 0 < d < 300 mm;

– exposure at higher distance 0 < d < 1 900 mm

For each distance d, the factor K is calculated for 5 different distances e between the 2 wires:

5 mm, 10 mm, 20 mm, 40 mm and 80 mm

For illustrations, three results of computation are presented, corresponding to 3 distances d

between the disk and the wires (d = 7,5 mm, 97,5 mm and 900 mm), and with e = 5 mm

NOTE d is the distance between the edge of the disk and the closest part of the source, i.e the closest wire

Considering the distance between the wires (e = 5 mm), a value d = 7,5 mm corresponds to a distance of 10 mm

between the edge of the disk and the median axis of the 2 wires

Standardised value of B at the

edge of the disk: 1,25 µT

Figure C.1 – Conductive disk in the magnetic field generated

by 2 parallel wires with balanced currents

C.1 Calculations for a conductive disk with a radius R = 100 mm

C.1.1 Examples of calculation of inducted currents in the disk

C.1.1.1 Distance to the source d = 7,5 mm

Results of the computation of local induced currents in the disk are given hereunder in form of graphs giving the shape of the distribution of induced currents in the disk (Figure C.2) The curve in Figure C.3 gives the numerical values of the distribution of the induced currents integrated over a surface of 1 cm2 perpendicular to the induced current direction

Figure C.2 – Current density lines J and distribution of J in the disk

(source: 2 parallel wires with balanced currents, separated by 5 mm,

located at d = 7,5 mm from the edge of the disk)

0 0,1 0,2 0,3

Figure C.3 – Ji = f [r]: Distribution of integrated induced current density calculated

along the diameter AA of the disk

(source: 2 parallel wires with balanced currents, separated by 5 mm,

located at d = 7,5 mm from the edge of the disk)

NOTE The diameter AA is located as illustrated in Figures C.1 and C.2

Diameter AA

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